fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...

, the pressure-gradient force is the

force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

that results when there is a difference in pressure across a surface. In general, a

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...

is a force per unit area, across a surface. A difference in pressure across a surface then implies a difference in force, which can result in an

acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...

according to

Newton's second law of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...

, if there is no additional force to balance it. The resulting force is always directed from the region of higher-pressure to the region of lower-pressure. When a

fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...

is in an

equilibrium state Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...

(i.e. there are no

net force Net Force may refer to: * Net force, the overall force acting on an object * ''NetForce'' (film), a 1999 American television film * Tom Clancy's Net Force, a novel series * Tom Clancy's Net Force Explorers Tom Clancy's Net Force Explorers or Ne ...

s, and no acceleration), the system is referred to as being in

hydrostatic equilibrium In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...

. In the case of

atmosphere An atmosphere () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A s ...

s, the pressure-gradient force is balanced by the

gravitational force In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...

, maintaining hydrostatic equilibrium. In

Earth's atmosphere The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing for ...

, for example,

air pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The Standard atmosphere (unit), standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equival ...

decreases at altitudes above Earth's surface, thus providing a pressure-gradient force which counteracts the force of gravity on the atmosphere.

Magnus effect

The

Magnus effect The Magnus effect is an observable phenomenon commonly associated with a spinning object moving through a fluid. The path of the spinning object is deflected in a manner not present when the object is not spinning. The deflection can be expl ...

is an observable

phenomenon A phenomenon ( : phenomena) is an observable event. The term came into its modern philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be directly observed. Kant was heavily influenced by Gottfried W ...

that is commonly associated with a spinning

object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...

moving through a

. The path of the spinning object is deflected in a manner that is not present when the object is not spinning. The deflection can be explained by the difference in pressure of the fluid on opposite sides of the spinning object. The Magnus effect is dependent on the speed of rotation.

Formalism

Consider a cubic parcel of fluid with a density

\rho

, a height

dz

, and a surface area

dA

. The mass of the parcel can be expressed as,

m = \rho \, dA \, dz

. Using Newton's second law,

F = m a

, we can then examine a pressure difference

dP

(assumed to be only in the

z

-direction) to find the resulting force,

F = - dP \, dA = \rho a \, dA \, dz

. The acceleration resulting from the pressure gradient is then,

a = -\frac \frac .

The effects of the pressure gradient are usually expressed in this way, in terms of an acceleration, instead of in terms of a force. We can express the acceleration more precisely, for a general pressure

P

as,

\vec = -\frac \vec\nabla P.

The direction of the resulting force (acceleration) is thus in the opposite direction of the most rapid increase of pressure.

References

* Roland B. Stull (2000) ''Meteorology for Scientists and Engineers'', Second Edition, Ed. Brooks/Cole, . Fluid dynamics Atmospheric dynamics {{climate-stub